# Ruling out limit cycles in 2 dimensions

Let $$f:[0,1]^2 \rightarrow \mathbb{R}^2,$$ where $$f_1(x,y) = g(y)-x$$ and $$f_2(x,y) = g(x)-y.$$ Here $$g(\cdot)$$ is a strictly decreasing polynomial function such that $$g(0)=1$$ and $$g(1)=0.$$ I am interested in analyzing the asymptotic behavior of the following system of differential equations:

$$$$\dot{x} = f_1(x,y) \ \text{and} \ \dot{y} = f_2(x,y).$$$$

I want to show that the above system's limiting behavior will be to one of the stationary states i.e., rule out limit cycles and other complicated behavior. Since $$\frac{\partial f_1}{\partial x} + \frac{\partial f_2}{\partial y} = -2 \neq 0,$$ can I use the Bendixson–Dulac theorem to rule out limit cycles and conclude that the above system converges to one of the stationary states?

The domain in the statement of Bendixson–Dulac's theorem is usually an open set. However my domain of interest $$[0,1]^2$$ is not open and so am confused. I would really appreciate help on this.

• I think you can still apply the theorem but just restrict the domain to $(0,1)^2$ Commented May 14, 2022 at 18:12
• @person Can I also conclude that the possible limiting behavior of the system is converging to one of the stationary states ? Commented May 15, 2022 at 7:05
• I believe so, but think you might have to be careful with trajectories starting on the border. Although since the equations are polynomials it should still work. Commented May 15, 2022 at 20:06